⟨a, b | aaa=1, abba=bbb⟩

Up:

Monoid enumeration

#8066 ⟨a, b | aaa=1, abba=bab⟩

#8071 ⟨a, b | aaa=1, abbb=abb⟩

Properties

Sum of relation sides is 10

Infinite non-cancellative non-commutative monoid

Element ba² has infinite order

Not left cancellative, because left multiplication by b is not injective:

b ⋅ b³¹ = b² and b ⋅ b = b², however b³¹ ≠ b

Not right cancellative, because right multiplication by b is not injective:

(ab)³ ⋅ b = bab³ab⁸ and bab³ab⁷ ⋅ b = bab³ab⁸, however (ab)³ ≠ bab³ab⁷

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b³² ⇒ b²	[25]
2.	b²ab³⁰ ⇒ b²a	[23]
3.	b⁵a ⇒ ab⁵	[5]
4.	a²b² ⇒ b²ab²⁹	[26]
5.	ab²a ⇒ b³	[2]
6.	ab⁴a ⇒ b³ab³	[6]
7.	b²a² ⇒ b⁴ab²⁷	[27]
8.	b(ba)² ⇒ (ab³)²	[10]
9.	(b³a)² ⇒ abab²	[11]
10.	a³ ⇒ 1	[1]
11.	a²bab² ⇒ b²ab³ab²⁷	[28]
12.	(ab)³b ⇒ bab³ab⁸	[13]
13.	(ba)²b³a ⇒ abab³ab¹⁹	[19]

# ab:aaa=1,abba=bbb b/a - - bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=bb bbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=bba bbbbba=abbbbb aabb=bbabbbbbbbbbbbbbbbbbbbbbbbbbbbbb abba=bbb abbbba=bbbabbb bbaa=bbbbabbbbbbbbbbbbbbbbbbbbbbbbbbb bbaba=abbbabbb bbbabbba=ababb aaa=1 aababb=bbabbbabbbbbbbbbbbbbbbbbbbbbbbbbbb abababb=babbbabbbbbbbb bababbba=ababbbabbbbbbbbbbbbbbbbbbb

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

Σ	#	Presentation	Mapping
11	22778	⟨a, b \| aaa=1, aabbb=bba⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

22778

⟨a, b | aaa=1, aabbb=bba⟩

φ(a) = a, φ(b) = b

#8067 ⟨a, b | aaa=1, abba=bbb⟩

Contents

Properties

Rewriting system

Isomorphic instances

Up:	Monoid enumeration
Prev:	#8066 ⟨a, b \| aaa=1, abba=bab⟩
Next:	#8071 ⟨a, b \| aaa=1, abbb=abb⟩